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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-10-05 22:44:03 UTC</tt>.<br>
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| : The original revision id was <tt>457120212</tt>.<br>
| | 578edo is [[enfactoring|enfactored]] in the [[5-limit]], [[tempering out]] [[32805/32768]] (schisma) with the same tuning as [[289edo]]. It tempers out 118098/117649 (stearnsma) which together with the schisma gives 7-limit [[pogo]] temperament, the 224 & 354 temperament. In the [[11-limit]] it tempers out [[540/539]] and [[4000/3993]] and provides the [[optimal patent val]] for 11-limit pogo and the planar temperament [[hades]], as well as other temperaments tempering out 540/539, the rank-4 temperament for which it also provides the optimal patent val. In the 13-limit, it tempers out [[729/728]], [[1575/1573]], [[1716/1715]] and [[2080/2079]], and provides the optimal patent val for 13-limit pogo. |
| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | === Prime harmonics === |
| <h4>Original Wikitext content:</h4>
| | {{Harmonics in equal|578|columns=11}} |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //578 equal division// divides the octave into 578 equal parts of 2.076 cents each. It is contorted in the 5-limit, tempering out 32805/32768 with the same tuning as 289edo. It tempers out 10976/10935 and 65625/65536 in the 7-limit, supporting the 22&118 temperament, and 119098/117649 which together with the schisma gives 7-limit [[Schismatic family#Pogo|pogo temperament]], the 94&130 temperament. In the 11-limit it tempers out 540/539 and 4000/3993 and provides the [[optimal patent val]] for 11-limit pogo and the planar temperament [[Swetismic temperaments#Hades|hades]]. In the 13-limit, it tempers out 729/729, 1575/1573, 1716/1715 and provides the optimal patent val for 13-limit pogo, as well as other temperaments tempering out 1716/1715, the rank five temperament for which it also provides the optimal patent val.</pre></div>
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| <h4>Original HTML content:</h4>
| | === Subsets and supersets === |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>578edo</title></head><body>The <em>578 equal division</em> divides the octave into 578 equal parts of 2.076 cents each. It is contorted in the 5-limit, tempering out 32805/32768 with the same tuning as 289edo. It tempers out 10976/10935 and 65625/65536 in the 7-limit, supporting the 22&amp;118 temperament, and 119098/117649 which together with the schisma gives 7-limit <a class="wiki_link" href="/Schismatic%20family#Pogo">pogo temperament</a>, the 94&amp;130 temperament. In the 11-limit it tempers out 540/539 and 4000/3993 and provides the <a class="wiki_link" href="/optimal%20patent%20val">optimal patent val</a> for 11-limit pogo and the planar temperament <a class="wiki_link" href="/Swetismic%20temperaments#Hades">hades</a>. In the 13-limit, it tempers out 729/729, 1575/1573, 1716/1715 and provides the optimal patent val for 13-limit pogo, as well as other temperaments tempering out 1716/1715, the rank five temperament for which it also provides the optimal patent val.</body></html></pre></div>
| | 578 factors as {{factorization|578}}, with divisors {{EDOs| 2, 17, 34, and 289 }}. |
| | |
| | [[Category:Swetismic]] |
| | [[Category:Hades]] |
| | [[Category:Pogo]] |
| Prime factorization
|
2 × 172
|
| Step size
|
2.07612 ¢
|
| Fifth
|
338\578 (701.73 ¢) (→ 169\289)
|
| Semitones (A1:m2)
|
54:44 (112.1 ¢ : 91.35 ¢)
|
| Consistency limit
|
7
|
| Distinct consistency limit
|
7
|
578 equal divisions of the octave (abbreviated 578edo or 578ed2), also called 578-tone equal temperament (578tet) or 578 equal temperament (578et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 578 equal parts of about 2.08 ¢ each. Each step represents a frequency ratio of 21/578, or the 578th root of 2.
578edo is enfactored in the 5-limit, tempering out 32805/32768 (schisma) with the same tuning as 289edo. It tempers out 118098/117649 (stearnsma) which together with the schisma gives 7-limit pogo temperament, the 224 & 354 temperament. In the 11-limit it tempers out 540/539 and 4000/3993 and provides the optimal patent val for 11-limit pogo and the planar temperament hades, as well as other temperaments tempering out 540/539, the rank-4 temperament for which it also provides the optimal patent val. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715 and 2080/2079, and provides the optimal patent val for 13-limit pogo.
Prime harmonics
Approximation of prime harmonics in 578edo
| Harmonic
|
2
|
3
|
5
|
7
|
11
|
13
|
17
|
19
|
23
|
29
|
31
|
| Error
|
Absolute (¢)
|
+0.000
|
-0.225
|
-0.155
|
+0.724
|
+0.931
|
+0.303
|
+0.927
|
-0.627
|
+0.791
|
+0.181
|
+0.985
|
| Relative (%)
|
+0.0
|
-10.8
|
-7.4
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+34.9
|
+44.9
|
+14.6
|
+44.6
|
-30.2
|
+38.1
|
+8.7
|
+47.5
|
Steps
(reduced)
|
578
(0)
|
916
(338)
|
1342
(186)
|
1623
(467)
|
2000
(266)
|
2139
(405)
|
2363
(51)
|
2455
(143)
|
2615
(303)
|
2808
(496)
|
2864
(552)
|
Subsets and supersets
578 factors as 2 × 172, with divisors 2, 17, 34, and 289.